Diophantine Geometry

seminar: Leaves of foliations with many rational points

speaker: Carlo Gasbarri (Université de Strasbourg)

abstract: Let $K$ be number field embedded in $\Bbb C$. Let $(X,\Cal F)$ be a foliated quasi projective variety defined over $K$. We will sketch the proof of the following statement: Let $A$ be an affine variety defined over $\Bbb C$ and $\gamma: A\to X(\Bbb C)$ be an analytic map such that i) $\gamma (A)$ is a leaf of the foliation; ii) $\gamma^{-1}(X(K))$ is Zariski dense in $A$; then the map $\gamma$ is algebraic. This generalizes a classical transcendence theorem by Bombieri and Schneider Lang. It implies for instance the following statement: Let $X$ be a quasiprojective variety defined over $K$ and $(E,\nabla)$ be a vector bundle equipped with an integrable connection over it. Suppose that $\sigma:X\to E$ is an analytic horizontal section of $(E,\nabla)$, then the restriction of $\sigma$ to the Zariski closure of $\sigma^{-1}(E(K))$ is algebraic.

timetable:
Tue 24 May, 11:00 - 12:00, Sala Conferenze Centro De Giorgi
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